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Entwicklung zu beteiligen! Dabei ist die Reihenfolge unerheblich und es wird nicht zurückgelegt. Die Definition ist wie folgt: \[\binom{n}{k} = \frac{n!}{k! \cdot (n - k)!} \] + \begin{satz} + Seien $k, n \in \N$. Dann gilt: + \[\binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}\] + \end{satz} \subsubsection{Binomialsatz}\label{kombi:binomsatz} Seien $x, y \in \R$ und $n \in \N$. Es gilt: \[{(x+y)}^n = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k} \] + \begin{satz} + \begin{enumerate} + \item \[\sum_{k=0}^n \binom{n}{k} = 2^n, \; \forall k,n \in \N_0\] + \item \[\sum_{k=0}^n {(-1)}^k \binom{n}{k} = \binom{n}{0} - \binom{n}{1} + + \binom{n}{2} \ldots = 0\] + \end{enumerate} + \end{satz} + \subsubsection{Siebformel}\label{kombi:sieb} Mithilfe der \idx{Siebformel} kann die Kardinalität einer Menge durch die Kardinalitäten ihrer Teilmengen bestimmt werden. @@ -80,7 +92,15 @@ Jeder ist dazu aufgerufen, sich an der Entwicklung zu beteiligen! Von $n$ Elementen gibt es genau $n!{}$ Permutationen, und ${(n-1)}!{}$ Permutationen mit dem Fixpunkt $k$. Die Anzahl der fixpunktfreien Permutationen ist - \[n!\frac{n!}{1!} + \frac{n!}{2} - \frac{n!}{3!} + \cdots + {(-1)}^n\frac{n!}{n!} = \sum_{k=0}^{n}{(-1)}^k\frac{n!}{n!} \] + \[n!\frac{n!}{1!} + \frac{n!}{2} - \frac{n!}{3!} + \cdots + {(-1)}^n\frac{n!}{n!} = \sum_{k=0}^{n} {(-1)}^k \frac{n!}{n!} \] + + \newcommand*{\fkn}{f: \{1, \ldots, k\} \rightarrow{} \{1, \ldots, n\}} + \begin{satz} + Die Anzahl der nicht-surjektiven Abbildungen $ \fkn$ ist gleich + \[\sum_{m=1}^k {(-1)}^{m-1} \binom{n}{m} (n-m). \] Die Anzahl + der Surjektionen beträgt + \[\sum_{m=0}^n {(=1)}^m \binom{n}{m} {(n-m)}^k \] + \end{satz} \subsubsection{Kombination mit Wiederholung}\label{kombi:mitWiederholung} Wenn die Reihenfolge egal ist und Wiederholungen erlaubt sind, wird @@ -102,10 +122,14 @@ Jeder ist dazu aufgerufen, sich an der Entwicklung zu beteiligen! \end{align*} Dann ist die Lösung: \begin{align*} - T(n) &= r^n T(0) + a \sum_{i=0}^{a-1} r^2, \; n \in \N_0, r = 1 \\ - T(n) &= r^a T(0) + a \frac{1-r^n}{1-r}, \text{\ für } r \ne 1 + T(n) &= r^n T(1) + a \sum_{i=0}^{a-1} r^2, \; n \in \N_0, r = 1 \\ + T(n) &= r^a T(1) + a \frac{1-r^n}{1-r}, \text{\ für } r \ne 1 \end{align*} + \subsubsection{Geometrische Summenformel} + Es gilt: + \[ \sum_{k=0}^n q^k = \frac{1-q^{n+1}}{1-q} \] + \subsection{Wachstum von Funktionen} Seien $f,g : \N_0 \rightarrow \R$.\\ \idx{Abschätzung nach oben}\\ @@ -154,7 +178,6 @@ Jeder ist dazu aufgerufen, sich an der Entwicklung zu beteiligen! Besitzt die Wurzel und jede innere Ecke genau $q$ direkte Nachfolger, ist dies ein \idx{vollständiger (n, q)-Baum} - \newtheorem{satz}{Satz}[section] \begin{satz} Sei $T$ ein (n, q)-Baum, wobei $n \ge 1, q \ge 2$. 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Dann ist $ (G \times H, 0) $ mit $ (g,h) (\in (G\times H)) \circ (g',h') (\in (G\times H)) = (g \cdot g', h*h') $ eine Gruppe + + \subsubsection{Identitätsfunktion} + TODO + + \subsubsection{Satz: Eigenschaften von Gruppen} + In jeder Gruppe gilt:\footnote{Beweise gab es in der VL, zu finden in den Notizen zu Gruppen im studIP} + \begin{enumerate}[a.)] + \item $ \exists ! e \in G $ (Gruppe enthält genau ein neutrales Element) + \item $ \forall a \in G $ gilt $ a * e = a $ + \item $ \forall a \in G, \exists! a' $ mit $ a' * a = e $ (es existieren alle inversen Elemente) + \item $ a' * a = e \Rarr a * a' = e $ + \item $ a * b = a * c \Rightarrow b = c $ und $ b * a = c * a \Rarr b = c $ + \end{enumerate} + \subsubsection{Definition: Untergruppen} + Eine Teilmenge $ U \subset G $ einer Gruppe $ (G, *) $ heißt \idx{Untergruppe} von G, wenn folgende Bedingungen erfüllt sind: + \begin{itemize} + \item $ U \neq \emptyset $ + \item $ a,b \in U \Rarr a * b \in U $ + \item $ a \in U \Rarr a^{-1} \in U $ + \end{itemize} + + \subsubsection{Definition: Gruppenhomomorphismus} + \textbf{i)} Eine Abbildung $ f:G\rarr H $ zwischen $ (G, *) $ und $ (H, \circ) $ heißt \idx{Gruppenhomomorphismus}, wenn $ \forall a,b \in G $ stets $ f(a*b) = f(a) \circ f(b) $ gilt.\\ + \textbf{ii)} Ein bijektiver Gruppenhomomorphismus heißt \idx{Isomorphismus}. Falls $ f: G \rarr H $ ein Isomorphismus ist, schreibt man $ f:G \tilde{\rarr} H$. + + \subsubsection{Definition: Zyklen} + Eine Gruppe heißt \idx{zyklisch}, wenn es ein $ g \in G $ gibt, sodass $ := \{g^k | k \in \Z \} = \{\dots,g^{-2}, g^{-1}, e, g^1, g^2, \dots\} $ + + \subsubsection{Satz: Untergruppe von $\Z$} + Zu jeder Untergruppe $ U \in \Z $ von $ (\Z,+) \exists $ ein $ n \in \Z $ mit $ U = n\Z$ + + + + \section{Vektorräume}\label{vektorraum} \subsection{Vektorraum}\label{vektorraum:def} Eine nichtleere Menge $V$ heißt Vektorraum über einem Körper $\K$, wenn die folgenden Eigenschaften zutreffen:\\ diff --git a/env/commands.tex b/env/commands.tex index f91970a..345bce6 100644 --- a/env/commands.tex +++ b/env/commands.tex @@ -39,3 +39,5 @@ % Index command to show the key emphasized \newcommand{\idx}[1]{{\emph{#1}\index{#1}}} + +\newtheorem{satz}{Satz}[section]